Efficiently computing (a - K) / (a + K) with improved accuracy

Question

In various contexts, for example for the argument reduction for mathematical functions, one needs to compute (a - K) / (a + K), where a is a positive v

Answer 1

If you can relax the API to return another variable that models the error, then the solution becomes much simpler:

float foo(float a, float k, float *res)
{
    float ret=(a-k)/(a+k);
    *res = fmaf(-ret,a+k,a-k)/(a+k);
    return ret;
}

This solution only handles truncation error of division, but does not handle the loss of precision of a+k and a-k.

To handle those errors, I think I need to use double precision, or bithack to use fixed point.

Test code is updated to artificially generate non zero least significant bits in the input

test code

https://ideone.com/bHxAg8